Rain in India

  • The climate of India predominately depends on rainfall
  • Average: 899 mm with variation \pm 20\%
  • Monsoon is the typical rainy season
    • Onset: May-July
    • Withdrawal: Sep-Oct
  • What is the Monsoon?

Average rainfall across India

The Monsoon System

Definition according to the Indian Meteorological Department (IMD)

The seasonal reversal of the direction of winds along the shores of the Indian Ocean, especially in the Arabian Sea, which blow from the southwest for half of the year and from the northeast for the other half.

  • Only partly understood
  • The most anticipated weather phenomenon
  • The biggest financial bet
  • Notoriously difficult to predict
  • Affecting the Indian flora and fauna, economy, and agriculture

Monsoon is Coming!

Onset: May–July

Withdrawal: Sep–Oct
  • The main objective of the project is develop an early warning system using topological data analysis (TDA).

Nuisances of Climate Data

  • high dimensionality
  • complexity of realistic models
  • presence of noise
  • missing values

Successful Applications of TDA

  • European Topsoil
    • Savic, Toth, and Duponchel (2017)
  • Wildfire
    • Kim and Vogel (2019)
  • Atmospheric River Patters
    • Muszynski et al. (2019)
  • Weather Regimes
    • Strommen et al. (2023)

Topology of a Time-series

Chaos

A chaotic system is deterministic but sensitive to initial conditions and manifest unpredictable patterns.

Lorenz System

\begin{aligned} \dot{x} &= \sigma(y-x)\\ \dot{y} &= x(\rho-z)-y\\ \dot{z} &= xy-\beta z \end{aligned} Parameters: \sigma = 10, \rho = 28, \beta = 8 / 3.

  • Chaos is still discernible when projected on the xy-plane.

  • How much of it still detectable using only a single signal, e.g. x?

Time-Delay Embedding

Let’s now consider only one signal: x

The topology of the time-series is shown below:

Sliding Window

We still consider only one signal: x

Below are the time-delay embeddings of the above windows:

TDA Pipeline

We use overlapping windows to detect transition to chaos.

  1. take the input signal

  2. set a window size and delay

  3. position the window at the beginning of the time-series, and do the following:

    • construct a point-cloud using time-delay embedding

    • compute persistence diagram

    • additionally, compute summaries like persistence landscapes, L^p norms

  4. slide the window one-step forward, and repeat 3 until the end of the time-series is reached.

  5. In our case, we accumulate the computed norms to output a time-series.

Predicting the Monsoon

The Monsoon Index

Q: Can you guess the official onset?

A: June 18 was the official date.

Inference using Sliding Window

Sliding Window: Window size 30, Delay 7

Onset and Withdrawal

Future Directions

References

Kim, Hannah, and Christian Vogel. 2019. “Deciphering Active Wildfires in the Southwestern USA Using Topological Data Analysis.” Climate 7 (12): 135. https://doi.org/10.3390/cli7120135.
Muszynski, Grzegorz, Karthik Kashinath, Vitaliy Kurlin, Michael Wehner, and Prabhat. 2019. “Topological Data Analysis and Machine Learning for Recognizing Atmospheric River Patterns in Large Climate Datasets.” Geoscientific Model Development 12 (2): 613–28. https://doi.org/10.5194/gmd-12-613-2019.
Savic, Aleksandar, Gergely Toth, and Ludovic Duponchel. 2017. “Topological Data Analysis (TDA) Applied to Reveal Pedogenetic Principles of European Topsoil System.” Science of The Total Environment 586 (May): 1091–1100. https://doi.org/10.1016/j.scitotenv.2017.02.095.
Strommen, Kristian, Matthew Chantry, Joshua Dorrington, and Nina Otter. 2023. “A Topological Perspective on Weather Regimes.” Climate Dynamics 60 (5-6): 1415–45. https://doi.org/10.1007/s00382-022-06395-x.